I need to prove the following theorem:
Let $V$ be a vector space over $F$. Let $v_1,v_2,...,v_r \in V$ and $r > dim(V)$. Then, $(v_1,v_2,...,v_r)$ is linearly dependent.
My Proof Attempt:
Let dim(V) = n, for convenience, and let $(u_1,u_2,...,u_n)$ be a basis for V. Clearly, $(v_1,v_2,....,v_r)$ cannot be a basis for V since r>n and all bases have the same length.
Now, all the vectors in $(v_1,v_2,...,v_r)$ can be written as linear combinations of the basis vectors, which generate V. Hence:
$L(v_1,v_2,...,v_r) = L(u_1,u_2,....,u_n) = V$
Since $(v_1,v_2,...,v_r)$ generates V but is not a basis, it follows that it cannot be linear independent. Hence, it is linearly dependent.
The book gives a proof that is based off of the Basis Extension Theorem and I understand that proof. I was just wondering if my approach is valid or not, since I attempted to prove the result before I looked at the one given by the book.